Constraints on sterile neutrinos as dark matter candidates
from the diffuse X-ray background
A. Boyarsky,
1,2A. Neronov,
3,4† O. Ruchayskiy
5and M. Shaposhnikov
1,2 1CERN, Theory department, Ch-1211 Geneve 23, Switzerland2Ecole Polytechnique F´ed´erale de Lausanne, Institute of Theoretical Physics, FSB/ITP/LPPC, BSP 720, CH-1015, Lausanne, Switzerland´ 3INTEGRAL Science Data Centre, Chemin d’ ´Ecogia 16, 1290 Versoix, Switzerland
4Geneva Observatory, 51 ch. des Maillettes, CH-1290 Sauverny, Switzerland 5Institut des Hautes ´Etudes Scientifiques, Bures-sur-Yvette, F-91440, France
Accepted 2006 April 15. Received 2006 April 5; in original form 2006 January 10
A B S T R A C T
Sterile neutrinos with masses in the kilo-electronvolt range are viable candidates for the warm dark matter. We analyse existing data for the extragalactic diffuse X-ray background for signa-tures of sterile neutrino decay. The absence of a detectable signal within current uncertainties of background measurements puts model-independent constraints on allowed values of the sterile neutrino mass and mixing angle, which we present in this work.
Key words: neutrinos – dark matter – X-rays: diffuse background.
1 I N T R O D U C T I O N
At the present time there exists an extensive body of evidence that most of the matter in the Universe is composed of new, as yet undis-covered particles – the dark matter (DM). Observations of (i) galactic rotation curves, (ii) cosmic microwave background radiation, (iii) gravitational lensing, and (iv) X-ray emission of hot gas in clusters of galaxies provide independent measurements of the DM content of the Universe.
A further major experimental discovery of the last decade is that of neutrino oscillations. There are separate measurements of neu-trino oscillations in solar neuneu-trinos (Ahmad et al. 2002), atmospheric neutrinos (Fukuda et al. 1998), and reactor neutrinos (Eguchi et al. 2003). Neutrino oscillations can be explained if a neutrino is a mas-sive particle, which is contrary to the Standard Model assumption. This means that, along with the usual left-handed (oractive)
neutri-nos, there may also exist right-handed (orsterile) neutrinos.
The conventional see-saw mechanism (Minkowski 1977; Yanagida 1979, 1980; Gell-Mann, Ramond & Slansky 1980; Ra-mond 1979; Mohapatra & Senjanovic 1980; Glashow 1980) for the generation of small active neutrino masses implies that the sterile neutrinos are heavy (usually of the order of the grand unification theory (GUT) energy scale,∼1010–1015GeV) and that their mixing
with usual matter is of the order of sinθ ∼ 10−10–10−15. In addi-tion to the smallness of neutrino masses, models of this type can explain the baryon asymmetry of the Universe via thermal lepto-genesis (Fukugita & Yanagida 1986) and anomalous electroweak
On leave of absence from Bogolyubov Institute of Theoretical Physics,
Kyiv, Ukraine.
†E-mail: andrii.neronov@obs.unige.ch
fermion number non-conservation (Kuzmin, Rubakov & Shaposh-nikov 1985). They do not, however, offer a DM candidate.
Recently it was proposed that neutrino oscillations, the origin of the dark matter, and the baryon asymmetry of the Universe could be consistently explained in a model termed the ‘neutrino Minimal Standard Model’ (νMSM) (Asaka, Blanchet & Shaposhnikov 2005; Asaka & Shaposhnikov 2005). This model is a natural extension of the Minimal Standard Model (MSM), in which three right-handed neutrinos are introduced into the MSM Lagrangian. In this exten-sion, neutrinos obtain Dirac masses via Yukawa coupling analogous to the other quarks and leptons of the MSM, and in addition Majo-rana mass terms are allowed for right-handed neutrinos. Unlike con-ventional see-saw scenarios, all of these Majorana masses (which are roughly equal to the masses of corresponding sterile neutrinos) are chosen such that the mass of the lightest sterile neutrino is in the kilo-electronvolt range, and the others are 100 GeV – below the electroweak symmetry-breaking scale. In this model, the role of the dark matter particle is played by the lightest sterile neutrino. The existence of a relatively light sterile neutrino has non-trivial observable consequences for cosmology and astrophysics. It was proposed in Dodelson & Widrow (1993) that a sterile neutrino with a mass in the kilo-electronvolt range may be a viable ‘warm’ DM candidate. The small mixing angle (sinθ ∼ 10−6–10−4) between sterile and active neutrinos ensures that sterile neutrinos were never in thermal equilibrium in the early Universe and thus allows their abundance to be smaller than the equilibrium one. Moreover, a sterile neutrino with these parameters is important for the physics of super-novas (Fryer & Kusenko 2006), and was proposed as an explanation of the pulsar kick velocities (Kusenko & Segre 1997; Fuller et al. 2003; Barkovich, D’Olivo & Montemayor 2004).
In addition to the dominant decay mode into three active neutri-nos, the light (with massms 1 MeV) sterile neutrino can decay
exists a possibility of direct detection of neutrino-decay emission lines from sources with a high concentration of DM, for exam-ple from clusters of galaxies (Abazajian et al. 2001). Similarly, the signal from radiative sterile neutrino decays accumulated over the history of the Universe could be seen as a feature in the diffuse extragalactic background light spectrum. This opens up the pos-sibility of studying the physics beyond the Standard Model using astrophysical observations.
Recently there have been a number of works devoted to the anal-ysis of the possibility of discovering sterile neutrino radiative decay from X-ray observations (Abazajian et al. 2001; Mapelli & Ferrara 2005). For example, it was argued by Abazajian et al. (2001) that, if sterile neutrinos made up 100 per cent of all the DM, one should be able to detect the DM decay line against the background of the X-ray emission from the Virgo cluster. According to Abazajian et al. (2001), the non-detection of the line puts an upper limit ofms <
5 keV on the neutrino mass [this limit was, however, recently revised in Abazajian (2006), who finds the restrictionms< 8 keV]. It was
also noted by Abazajian et al. (2001) and Mapelli & Ferrara (2005) that one can obtain even stronger constraints,ms 2 keV, from the
diffuse extragalactic X-ray background (XRB) under the assump-tion that the dark matter in the Universe is uniformly distributed up to distances corresponding to redshiftsz 1. Together with the
claims of Hansen et al. (2002) and Viel et al. (2005), putting a lower bound ofms> 2 keV on the neutrino mass from Lyman α forest
observations, this would lead to a very narrow window for allowed sterile neutrino masses, if not excluding it completely.
In this paper we re-analyse the limit imposed on the parameters of sterile neutrinos from the observations of the diffuse XRB. In order to do this, we process actual astrophysical data fromHEAO-1
andXMM–Newton missions. There are several motivations for this.
(i) All the above restrictions on sterile neutrino mass (Abaza-jian et al. 2001; Dolgov & Hansen 2002; Mapelli & Ferrara 2005; Abazajian 2006) are model-dependent and based on the assumption that sterile neutrinos were absent in the early Universe at tempera-tures greater than a few giga-electronvolts. Depending on the model, the relationship between the mass of sterile neutrino, the mixing an-gle and the present-day sterile neutrino density, s, changes. In fact, to compute the sterile neutrino abundance one needs to know whether there was any substantial lepton asymmetry of the Uni-verse at the time of sterile neutrino production, what the coupling of sterile neutrinos to other particles such as inflatons or super-symmetric particles is, etc.1 Moreover, even if these uncertainties
were removed, a reliable computation of the relic abundance of ster-ile neutrinos is very difficult as the peak of their production falls on the Quantum Chromodynamics (QCD) epoch of the Universe evo-lution, corresponding to the temperature∼150 MeV (Dodelson & Widrow 1993), where neither a quark–gluon nor hadronic descrip-tion of the plasma is possible. Therefore, until the particle physics model is fully specified and the physics of a hadronic plasma is fully understood, one cannot put a robust restriction on one single parameter, such asms, of the model.
Therefore we aim in this paper at a clear separation between the model-independent predictions, based solely on astrophysical observations, and any statements that depend on a given model and underlying assumptions. To this end, we treatmsand sinθ as two
1For example, if the coupling of sterile neutrinos to inflatons is large enough, the main production mechanism will be the creation of sterile neutrinos in inflaton oscillations rather than active–sterile neutrino transition.
independent parameters, and present the limits in the form of an
‘exclusion plot’ in the (ms;ssin22θ) parameter space.
It should be stressed that our data analysis is not based on any specific model of sterile neutrinos, and as such can be applied to any ‘warm’ DM candidate particle that has a radiative decay channel. In the case of sterile neutrinos, the full decay width of this process is related to the parameter sinθ via equation (4) (see below).
(ii) In contrast to previous works (Abazajian et al. 2001; Mapelli & Ferrara 2005), we argue that the non-isotropy or ‘clumpyness’ of the matter distribution in the nearby Universedoes not relax the
limit on the neutrino mass. Indeed, the fact that a significant part of the dark matter at redshiftsz 10 is concentrated in galaxies
and clusters of galaxies just means that the strongest signal from the dark matter decay should come from the sum of the signals from the compact sources atz 10. Taking into account that the DM decay
signal fromz 10 is some two orders of magnitude stronger than
that fromz 10, while the subtraction of resolved sources reduces
the residual X-ray background maximum by a factor of 10, we argue that it would be wrong to subtract the contribution from the resolved sources from the XRB observations when looking for the DM decay signal. The form of the XRB spectrum with sources subtracted is, in fact, unknown, and the assumption that it has the shape of the initial spectrum, scaled down according to the resolved fraction (as in Abazajian et al. 2001), requires additional justification.
(iii) We find that a more elaborate analysis of the data enables us to put tighter limits on the allowed region of the parameter space (ms,ssin22θ) from the XRB observations. The idea is that
the cosmological DM decay spectrum is characterized not only by the total flux but also by a characteristic shape. Being present in the XRB spectrum, it would produce a local feature with some clear maximum and a width greater than the spectral resolution of the instrument. Features of such a scale are clearly absent in the data (smaller features could be present in the spectrum as a result of, for example, element lines, but they cannot produce a bump wider than the spectral resolution). Therefore, one can find that the addition to the standard broad continuum model of the XRB of the DM decay component in the wrong place results in a decrease of the overall quality of the fit of the data with such a two-component model (i.e. an increase in theχ2of the fit). The condition that the two-component
model provides an acceptable fit to the data imposes an upper limit on the flux in the DM decay component that is much more restrictive than the limit following from the condition that the flux of the DM component should not exceed the flux in the continuum component. The paper is organized as follows. In Section 2 we compute the contribution of the radiative decay of sterile neutrinos to the diffuse XRB and compare its shape with that of the measured XRB. We discuss the effects of the non-uniformness of the DM distribution in Section 2.1. In Section 3 we obtain a model-independent exclusion region fromHEAO-1 and XMM–Newton observations.
2 T H E C O N T R I B U T I O N O F D M D E C AY S T O T H E X R B
A sterile neutrino with a decay width and mass msdecays into an active neutrino and emits photons with a line-like spectrum at the energyEγ= ms/2. However, photons emitted at different
cosmo-logical distances are redshifted on their way to the Earth, so that as a result the photon spectrum is given by
d2N d dE = n0 DM 4π 1 E H (ms/(2E) − 1) (1)
(see Masso & Toldra 1999; Abazajian et al. 2001). Here,n0 DMis the
DM number density at the present time, andH(z) is the Hubble
pa-rameter as a function of redshift. The explicit form ofH(z) depends
on the cosmological parameters. This means that the expected dark matter decay spectrum is different for different cosmologies. We are interested inz corresponding to the epoch more recent than the
radiation-dominated Universe, and, therefore,H(z) can be written
as
H (z) H0
+ matter(1+ z)3, (2)
whereH0is the present-day value of the Hubble parameter and,
matterare the cosmological constant and matter contributions to the density of the Universe. Substituting (2) into (1), we find that for smallz the spectrum is approximated as
d2N d dE n0 DM 2πH0 (2E)1/2 8E3 + matterm3s . (3)
Assuming the Majorana nature of neutrino mass, the decay width is related to sin 2θ via (Pal & Wolfenstein 1982; Barger, Phillips & Sarkar 1995) = 9α G2F 256× 4π4 sin 2(2θ) m5 s = 5.6 × 10−22sin2θ m s 1 keV 5 s−1, whereGFis the Fermi constant. (4)
As an example, we show in Fig. 1 the expected dark matter decay spectrum for the cosmological ‘concordance’ model with 0.7,
matter 0.3. One can see that close to the maximum the spectrum is
roughly a power law with photon index equal to 1 (dN/dE ∼E−1).
Figure 1. Diffuse X-ray background spectrum from the HEAO-1 mission. The black solid curve shows the empirical fit equation (5) by Gruber et al. (1999). The reducedχ2of this fit is 1.2. Dashed (green and red) lines rep-resent the result of the fit of the same data to a model of the form (5) with an added DM component. The DM decay component (dot–dashed curve) is calculated for the ‘concordance’ model= 0.7, matter= 0.3 and for ms = 36.5 keV. The green (long-dashed) line represents a fit of DM with the mixing angle sin22θ = 1.9 × 10−12, and the best achievable fit has the
re-ducedχ2= 2. For the red (short-dashed) line, the mixing angle is sin22θ = 2.4× 10−13and the reducedχ2= 3, which we choose as a border line for the allowed quality of fit.
The measurements of the diffuse XRB (Marshal et al. 1980; Gruber et al. 1999) show that for 3 E 60 keV its form is well approximated by the following analytical expression:
d2F XRB dE d = CXRBexp − E TXRB E 60 keV − XRB+1 keV keV sr s cm2, (5)
where CXRB is a normalization constant. In Fig. 1 we show the
HEAO-1 data points2 together with the above analytical fit (solid
black line). Such a form of the XRB spectrum can be explained by active galactic nucleus (AGN) emission, under certain assumptions about AGN populations (Worsley et al. 2005; Treister & Urry 2005). Below∼15 keV, the XRB was measured by many X-ray missions and the background was found to follow a power law with pho-ton index XRB 1.3–1.4 (Gruber et al. 1999; Lumb et al. 2002; Revnivtsev et al. 2003; Revnivtsev et al. 2005; Gilli 2003; Barger 2003). The analysis of Gruber et al. (1999) finds TXRB = 41.13
keV andCXRB= 7.9. The DM decay component produces a harder
spectrum, as can be seen from Fig. 1.
2.1 The uniformness of the DM density in the Universe
Most of the power in the very hard DM decay spectrum of Fig. 1 is emitted in the narrow energy interval close to the maximal energy
Emax ms/2. From equation (3) it can be seen that emission in
this energy range is produced by neutrinos decaying at the present epoch (z 0). For the case of the ‘concordance’ model, the DM
decay spectrum is characterized by the very hard inverted power law dN/dE ∼ E1/2below an energy of∼Emax/2. The energy flux drops
by an order of magnitude at energies∼Emax/4, which correspond to the redshiftz 3. Thus, most of the DM decay signal is collected
from low redshifts.
It is known that the process of structure formation leads to sig-nificant clustering of the dark matter at small redshifts. The signal from DM decays at smallz is thus dominated by the sum of
con-tributions from the point-like sources corresponding to large DM concentrations, such as galaxies or clusters of galaxies.
Abazajian et al. (2001) and Mapelli & Ferrara (2005) argued that the clustering of the DM at small redshifts makes if difficult for the instruments that measure the diffuse XRB to detect the signal from the DM decays at small redshifts. Indeed, measurements of the diffuse XRB carried out with narrow-field instruments, such as
Chandra or XMM–Newton, can miss the DM signal because of the
absence of large nearby galaxies or clusters of galaxies in the fields used for the deep observations and background measurements. As a measure for galaxy clustering, one can take the distribution of the number of galaxies as a function of the distanceNgal(r ). The Sloan
Digital Sky Survey data show that the functionNgal(r ) becomes
constant forr 100 Mpc (which corresponds to a redshift z ∼
0.02) (see, for example, Joyce et al. 2005). Therefore, an instrument with a field of view (FoV) that encompasses a volume∼(100 Mpc)3
at distances corresponding toz 1 will observe a homogeneous
matter distribution. Let us take the Hubble distance as an estimate for such distances:H−10 ∼ 3.8 × 103Mpc (here,H
0is the
present-day Hubble constant). Then, the minimal FoVθfovis determined
from a simple relationship,θ2
fovH−30 (100 Mpc)3, i.e.θfov 15.
The FoV ofXMM–Newton is precisely of this order, namelyθxmm∼
30. Therefore, the strongest signal from smallz can be missed if only
‘empty fields’ are selected for theXMM–Newton XRB observations.
2We thank D. Gruber for sharing these data with us and for many useful comments.
On the other hand, wide FoV instruments or the instruments that have conducted all-sky surveys, such asHEAO-1 or ROSAT, cannot
miss the largest contribution to the DM decay signal fromz< 10
because of the full sky coverage. Thus, the above argument should be used in a reverse sense: to find the DM decay signal it is necessary to use the data on the XRB collected from the whole sky, rather than from the ‘deep field’ observations of a narrow-field instrument.
3 R E S T R I C T I O N S O N PA R A M E T E R S O F S T E R I L E N E U T R I N O S F R O M M E A S U R E M E N T S O F T H E D I F F U S E X - R AY B AC K G R O U N D
3.1 Restrictions from HEAO-1 measurements
The above analytical approximation (5) provides a good fit to the
HEAO-1 data. No DM decay feature with a spectrum of the form
of (3) (corrected for the spectral resolution of the instrument) is evident in the data. A straightforward constraint on the possible contribution of DM decays to the XRB spectrum is that the flux of the DM decay contribution does not exceed the total flux. Such an approach was used by Dolgov & Hansen (2002), who applied it to the broad-band XRB model of Ressel & Turner (1989). Using the fit (5) of Gruber et al. (1999), which gives a much better description in the kilo-electronvolt region, we obtain an exclusion curve, shown as the green dot–dashed line in Fig. 2.
Clearly this restriction is model-independent and rigorous. How-ever, it can be made stronger. Indeed, the experimentally mea-sured XRB is monotonic at energies from kilo-electronvolts to giga-electronvolts (see Gruber et al. 1999). If the flux of the DM had
Figure 2. Exclusion plot on parameters msand sin22θ using HEAO-1 and
XMM–Newton data. The values in the non-shaded region are allowed. In the region where bothHEAO-1 and XMM–Newton data are available, HEAO-1 provides a more stringent constraint (as discussed in Section 3.2). We as-sumed that sterile neutrinos constitute 100 per cent of all the DM (i.e.
s = DM). To remove dependence on the value ofDM, we choose to plotssin22θ rather than sin22θ. The results can be described by a sim-ple empirical formula, equation (6) (thin blue dashed line). The dot–dashed green line represents the exclusion region obtained if all 100 per cent of the XRB flux is attributed to DM at energiesE ms/2.
contributed a significant part to the total XRB, contributions to the XRBunrelated to the radiative DM decay should combine into a
spectrum with a sudden narrow ‘dip’. In particular, at energiesE ms/2 some new physical phenomena should have suddenly ‘kicked
in’, causing the spectrum to experience a very sharp ‘jump’. More-over, the form of this dip must have been almost the same as the DM contribution with a minus sign. This would have required a mechanism for a very precise fine-tuning of contributions to the XRB between the DM and various physical phenomena, or simply a chance coincidence. Of course, such a conspiracy cannot be ab-solutely excluded, in particular because of the lack of unambiguous theoretical predictions of the shape of the XRB spectrum. However, it is very unlikely that there would be a precise cancellation of con-tributions of a different nature. Therefore, in what follows we will assume that such a situation does not occur (in particular, this does not occur in models attributing the existing XRB shape to AGN emissions: Worsley et al. 2005; Treister & Urry 2005).
In the present work we therefore argue that the constraint on the possible DM contribution to the XRB resulting fromHEAO-1
data can be improved using a statistical analysis of the data. Our strategy will be the following. We take the actualHEAO-1 data and
fit them to a model of the form (5) (varyingCXRB,TXRB, XRB)
plus an additional DM flux (3) (corrected for the spectral resolution ofHEAO-1 instruments). The addition of a large DM contribution
would worsen the quality of the model fit to the data, while the addition of a small DM contribution does not change the overall fit quality. Thus, one can put a restriction on the DM contribution by allowing the DM component to worsen the fit by a certain value. Taking into account that we have around 40 degrees of freedom of the system under consideration, we take the maximal allowed value of the reduced chi-square of the fit to beχ2< 3. Thus on a technical
level our method restricts the flux of DM to be of the order of errors of measured XRB flux, rather than its total value.
The results of the application of the above algorithm to the data are shown in Fig. 2. The allowed values of (ms,ssin22θ) are those in
the unshaded region.3The shaded region below the dot–dashed line
is excluded under the assumption that the XRB spectrum does not have an unknown feature (a ‘dip’) fine-tuned to be located precisely at energies at which DM contributes.
According to equations (3) and (4), the flux of DM is propor-tional tos sin2 2θ. We therefore choose to plot this value rather
than the conventional sin22θ on the y-axis of Fig. 2. This removes
uncertainty, related to the determination ofDM. Notice that the results obtained fromHEAO-1 data (thick solid lines in Fig. 2) are
model-independent and can be applied to any DM candidate that has a decay channel into a lighter particle and a photon, with its decay width related to sin θ via equation (4).
3.2 Restriction from XMM–Newton background measurements
In the energy band below 10 keV the XRB has been studied by numerous narrow-FoV instruments, includingXMM–Newton. The
increased angular resolution and sensitivity of these instruments
3The normalization of the XRB spectrum, measured byHEAO-1, is known to be lower than any other XRB measurements (Gilli 2003; Moretti et al. 2003). In Fig. 2, therefore,HEAO-1 data were increased by 40 per cent according to De Luca & Molendi (2005), Worsley et al. (2005) and Treister & Urry (2005) (see, however, Gilli 2003). This weakens the restriction of Fig. 2 by about 10 per cent, as compared with actualHEAO-1 data.
has enabled some 90 per cent of the XRB to be resolved into point sources (Brandt & Hasinger 2005; De Luca & Molendi 2005; Bauer et al. 2004). In theory, better sensitivity should enable us to put tighter constraints on the possible DM contribution to the XRB. However, in this section we show that this is not the case.
The key point that leads to such a conclusion is that the better sensitivity (to point sources) of these instruments is achieved as a result of the better angular resolution, rather than as a result of the increase of the effective collection area. Thus, if one is interested in the diffuse sources, the sensitivity is not improved compared with
HEAO-1. On the contrary, in the narrow-FoV instruments the XRB
signal is collected from a smaller portion of the sky, which leads to lower statistics of the signal. At the same time, the instrumen-tal background (which is thought to arise from cosmic rays hitting the instrument) is roughly the same for narrow- and wide-field in-struments. Thus, the ratio of the counts arising from the photons of the XRB to the instrumental background counts is smaller for the narrow-field instruments, and one needs larger integration times to achieve good statistical significance of the XRB signal.
The above problem would not affect the constraints onms, sinθ
if they were imposed by the condition for the flux in the DM de-cay component not to exceed the total XRB flux in a given energy interval. However, as the statistical significance of the XRB signal in the narrow-FoV instruments is lower, bigger errors decrease the
χ2-value, and therefore the imposed limit on DM will be worse than
that of the corresponding wide-FoV instruments.
The exclusion plot in the (ms,ssin22θ) parameter space
ob-tained from the analysis of the XMM–Newton data is shown in
Fig. 2. We took the actual data of two collections ofXMM–Newton
background observations (Lumb et al. 2002), total exposure time
∼450 ks; and Read & Ponman (2003), total observation time ∼1 Ms. The form of the XMM–Newton background is fitted by
a power law with index = 1.4 (Lumb et al. 2002)), in agreement with equation (5). Applying our method toXMM–Newton data, as
described in the previous section, we find the restrictions shown in Fig. 2. In the neutrino mass region 6 ms 20 keV, where
bothHEAO-1 and XMM–Newton data are available, the XMM– Newton background provides a weaker restriction than the data from HEAO-1, because of the reason discussed above (the XMM–Newton
FoV has a radius of 15 arcmin, which is much smaller than that of theHEAO-1, namely∼3◦ for A2 HED detectors). In the case
ofXMM–Newton, the statistical error at energies above roughly
7 keV is dominated by instrumental background, while in the case ofHEAO-1 the errors at these energies are dominated by the
dif-fuse XRB. As a result, errors of flux determination are smaller for HEAO data, thus providing more stringent restrictions. Of course, theXMM–Newton data provide constraints on the parameters of the
neutrino in the region 1 ms 6 keV, where HEAO-1 data are not
available. The data in Fig. 2 fit to the simple empirical formula
ssin2(2θ) < 3 × 10−5 ms keV −5 . (6) 3.3 Discussion
In this work we have looked for signatures of sterile neutrino decay in the extragalactic XRB. The main result of our paper is the plot in Fig. 2, which constrains the properties of sterile neutrinos as dark matter candidates.
Let us compare our exclusion plot with similar plots found in other works. Dolgov & Hansen (2002) used a broad-band limit, put on the XRB by Ressel & Turner (1989), to find restrictions on sterile
neutrino parameters. First of all, in the kilo-electronvolt energy band the limit of Ressel & Turner (1989) is weaker than that of Gruber et al. (1999), which we use in this paper. Therefore the dot–dashed line in Fig. 2 provides model-independent restrictions on the param-eters of sterile neutrinos that are stronger than the similar bound in fig. 4 of Dolgov & Hansen (2002). Second, as discussed in Section 3, we put a restriction on the parameters of sterile neutrinos based on the statistical analysis of the actual experimental data ofHEAO-1
and XMM–Newton missions. Such a constraint is possible under
the assumption that there is no extremely unlikely fine-tuning be-tween various components contributing to the XRB. In other words, the observed XRB spectrum is monotonic in the kilo-electronvolt range (at energies below ∼15 keV this was checked by various X-ray missions, including recent measurements byXMM–Newton
andChandra). Therefore, we assume that the XRB spectrum with-out DM has no ‘dip’ and exactly matches the shape and location
of the DM contribution. The results of our analysis are shown in the thick black solid line in Fig. 2. As the statistical qual-ity of the data is good, this gives about two orders of magni-tude stronger restrictions than those represented by the dot–dashed line. The region between the solid (or its empirical fit, equation 6) and dot–dashed lines is excluded under the assumptions discussed above.
Abazajian et al. (2001) provide an exclusion plot (also quoted by Abazajian 2006) based on observations of the Virgo cluster. Boyarsky et al. (2006) show that the restrictions from clusters of galaxies are weaker than those of Abazajian (2006); however, in the mass range 2 keV ms 10 keV they are 2 to 4 times better
than those coming from the XRB. For a detailed comparison of our results with those of Abazajian (2006), see Boyarsky et al. (2006).
In some cases, such as in the Dodelson–Widrow (DW) scenario, (Dodelson & Widrow 1993)4 there exists a relationship between
the mass of a sterile neutrino, the mixing angle and the present-day sterile neutrino densitys (Dolgov & Hansen 2002; Abaza-jian et al. 2001; AbazaAbaza-jian 2006). Because both the sterile neutrino abundance and the gamma-ray flux depend on the mixing angle
θ and the sterile neutrino mass ms only, this provides an upper limit on the sterile neutrino mass in the DW scenario. According to the computation of sterile neutrino abundance made in Abazajian (2006), the corresponding limit following from the X-ray bound found in the present paper readsms < 8.9 keV for the DW sterile
neutrino. However, even in the DW scenario this number is subject to uncertainties that are difficult to estimate, because the peak of the sterile neutrino production falls into the range of temperatures where neither a hadronic nor a quark description of the hot plasma is possible.
The Standard Model augmented by just one sterile neutrino can-not explain the neutrino oscillation data, and thus extra ingredi-ents should be added to it. The analysis of Asaka, Kusenko & Shaposhnikov (2006) shows that, in a more realistic model, the
νMSM, the relationship between the mass of the sterile neutrino,
the mixing angle and the present-day sterile neutrino density s changes, depending on unknown parameters, such as the masses and couplings of extra particles and primordial abundances of ster-ile neutrinos. Therefore, for a general case one cannot put a model-independent restriction on the mass of the sterile neutrinoms, and
4In this scenario it is assumed that there is just one species of sterile neutrino, that their concentration was zero at temperatures higher than 1 GeV, that there were no substantial lepton asymmetries, and that the Universe reheating temperature was larger than 1 GeV.
the search for a narrow line coming from its decays should be carried out in all energy ranges.
The results of this work show that a possible strategy to look for a feature of DM decay in the XRB is to use data from ‘all-sky surveys’ rather than ‘deep-field observations’, and to utilize instruments with the largest possible FoV (rather than the best angular/spectral res-olution). An example of such an instrument is INTEGRAL, with
which observations of the XRB are currently planned.
AC K N OW L E D G M E N T S
We would like to acknowledge useful discussions with K. Abazajian, A. Kusenko, I. Tkachev and R. Rosner. This work was supported in part by the Swiss Science Foundation and by European Research Training Network contract 005104 ‘ForcesUniverse’. OR was sup-ported by a Marie Curie International Fellowship within the 6th European Community Framework Programme.
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